Reducing fractions to their lowest terms

Step  1  :

            9
 Simplify   —
            a

Equation at the end of step  1  :

        9     
  (a -  —) -  4
        a     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  a  as the denominator :

          a     a • a
     a =  —  =  —————
          1       a  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 a • a - (9)     a2 - 9
 ———————————  =  ——————
      a            a   

Equation at the end of step  2  :

  (a2 - 9)    
  ———————— -  4
     a        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  a  as the denominator :

         4     4 • a
    4 =  —  =  —————
         1       a  

Trying to factor as a Difference of Squares :

 3.2      Factoring:  a² - 9 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 9 is the square of 3
Check :  a²  is the square of  a¹ 

Factorization is :       (a + 3)  •  (a - 3) 

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (a+3) • (a-3) - (4 • a)     a2 - 4a - 9
 ———————————————————————  =  ———————————
            a                     a     

Trying to factor by splitting the middle term

 3.4     Factoring  a² - 4a - 9 

The first term is,  a²  its coefficient is  1 .
The middle term is,  -4a  its coefficient is  -4 .
The last term, "the constant", is  -9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -9 = -9 

Step-2 : Find two factors of  -9  whose sum equals the coefficient of the middle term, which is   -4 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  a2 - 4a - 9
  ———————————
       a